Question

"Which algorithm efficiently finds the shortest path in a weighted graph with both positive and negative edge weights?

A. Dijkstra's algorithm
B. Breadth-First Search (BFS)
C. Bellman-Ford algorithm
D. A* algorithm"

Answer 1

The Bellman-Ford algorithm is the efficient algorithm that finds the shortest path in a weighted graph with both positive and negative edge weights.

Step-by-step explanation:

The algorithm that efficiently finds the shortest path in a weighted graph with both positive and negative edge weights is the Bellman-Ford algorithm. It is named after its developers, Richard Bellman and Lester Ford Jr. This algorithm can handle negative edge weights, which sets it apart from Dijkstra's algorithm.

The Bellman-Ford algorithm works by repeatedly relaxing the edges of the graph until it finds the shortest paths from the source vertex to all other vertices. It keeps track of the shortest distances and updates them if a shorter path is found. If the algorithm detects a negative cycle, it indicates that the graph contains no shortest paths.

Here is a step-by-step explanation of the Bellman-Ford algorithm:

1. Initialize the distances from the source vertex to all other vertices as infinity, except for the source vertex itself, which is set to 0.
2. Iterate through all the edges of the graph and relax each edge. Relaxing an edge means updating the distance of the destination vertex if a shorter path is found.
3. Repeat step 2 for a total of V-1 iterations, where V is the number of vertices in the graph. Each iteration guarantees that the algorithm finds the shortest paths consisting of at most i+1 edges, where i is the current iteration index.
4. After the V-1 iterations, check for negative cycles. If a shorter path is still possible, then the graph contains a negative cycle, and the Bellman-Ford algorithm cannot find the shortest paths.